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Orienteering with One Endomorphism.

Sarah Arpin1, Mingjie Chen2, Kristin E Lauter3

  • 1Mathematics Institute, Universiteit Leiden, Leiden, The Netherlands.

La Matematica
|October 2, 2023
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Summary

This study explores path-finding in supersingular isogeny cryptography by analyzing endomorphisms. New algorithms leverage the volcano structure for general path-finding, even without knowing the primitive order.

Keywords:
Elliptic curveEndomorphism ringOrientationPath-findingSupersingular isogeny graphVectorization

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Area of Science:

  • Number Theory
  • Cryptography
  • Post-Quantum Cryptography

Background:

  • Supersingular isogeny-based cryptography relies on solving the path-finding problem, often linked to the endomorphism ring problem.
  • A known endomorphism can simplify path-finding and endomorphism ring computation in supersingular elliptic curves.

Purpose of the Study:

  • To investigate if path-finding can be achieved using a single endomorphism.
  • To develop general path-finding algorithms for supersingular isogeny graphs without assuming knowledge of the primitive order.

Main Methods:

  • Utilizing the volcano structure of oriented supersingular isogeny graphs to perform graph traversal (ascending/descending/horizontal steps).
  • Introducing and solving a new hard problem: computing the primitive order from an endomorphism.
  • Developing a sub-exponential quantum algorithm for the new primitive order computation problem.

Main Results:

  • Path-finding algorithms are deduced by navigating the volcano structure of supersingular isogeny graphs.
  • A new computationally hard problem related to primitive orders and endomorphisms is introduced.
  • A quantum algorithm with sub-exponential complexity is presented for solving this new problem.

Conclusions:

  • The proposed path-finding algorithms are more general than prior methods as they do not require prior knowledge of the primitive order.
  • The research contributes to understanding the relationship between endomorphisms, primitive orders, and path-finding in supersingular isogeny cryptography.
  • The introduction of a new hard problem and a quantum algorithm advances the study of isogeny-based cryptosystems.